2014 AMC 10A Problem 21

Note that the lines intersect the $x$-axis when $y = 0.$ This gives us $$0 = ax + 5$$ and $$0 = 3x + b,$$ which when solved gives us $$x = -\dfrac{5}{a}$$ and $$x = -\dfrac{b}{3}.$$ Setting these equal to each other, we have $$\dfrac{5}{a} = \dfrac{b}{3}$$$$ab = 15.$$ We know that $a$ and $b$ are positive, which means that the only pairs of values $(a, b)$ that satisfy the above equation are $$(1, 15),$$$$(3, 5),$$$$(5, 3),$$ $$(15, 1).$$ Plugging these values back into the equations gives us $x$-values of $$x = -5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}.$$ The sum of all these values is $-8.$ Thus, E is the correct answer.